Preprint Number ??–??
ON LINEAR SPECTRAL TRANSFORMATIONS AND THE LAGUERRE-HAHN CLASS
K. CASTILLO AND M.N. REBOCHO
Abstract: We study the Christoffel, Geronimus, and Uvarov transformations for Laguerre-Hahn orthogonal polynomials on the real line. It is analysed the modi-fication of the corresponding difference-differential equations that characterize the systems of orthogonal polynomials and the consequences for the three-term recur-rence relation coefficients.
Keywords: Orthogonal polynomials; Laguerre-Hahn class; Riccati differential equa-tion; Christoffel transformaequa-tion; Geronimus transformaequa-tion; Uvarov transformation. Math. Subject Classification (2000): 33C47, 42C05.
1. Introduction
Spectral transformations of orthogonal polynomials is a widely known theme in the literature of special functions and applications (see, for instance, [11,16,17,21] and references therein). The canonical linear spectral transfor-mations are related to the Christoffel, Geronimus, and Uvarov modifications, that is, the modification of the orthogonality measure by a polynomial, a rational function, and the addiction of a mass point, respectively. A basic topic of research concerns the modification of real weights and the formulae expressing the new orthogonal polynomials in terms of the original system. The seminal references on this topic are traced back to [5,9,10,19,20]. More recent literature can be found, amongst many others, in [3, 16, 17, 21, 22].
The main motivation for the present work relies on studies concerning the Christoffel, Geronimus, and Uvarov transformations related to semi-classical measures. The semi-classical character is preserved under such transforma-tions [22]. Basic structures, such as the recurrence relation for the orthogonal polynomials, the spectral derivatives of the system, and the class of the cor-responding linear functional, have been extensively studied (see [3, 16, 21]
Received September 29, 2016.
KC is supported by the Portuguese Government through the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) under the grant SFRH/BPD/101139/2014. This work is partially supported by the Centre for Mathematics of the University of Coimbra UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.
and its references). Generalizations and extensions of semi-classical fam-ilies of orthogonal polynomials are well-known, their study can be placed within the so-called Laguerre-Hahn families. Similarly to the semi-classical case, the Laguerre-Hahn class is also closed under the Christoffel, Geronimus, and Uvarov transformations [6, 8, 13, 14]. A systematic study of Laguerre-Hahn orthogonal polynomials on the real line was initiated in [7], showing some basic properties such as the so-called structure relations, that is, sys-tems of difference-differential equations involving the orthogonal polynomi-als, {Pn}n≥0, and the associated polynomials, {P
(1)
n }n≥0 (cf. Section 2). In
the matrix form, these systems are given by
AΨ0n = MnΨn+ NnΨn−1, n ≥ 0 , Ψn =
h
Pn+1 Pn(1)
iT
. (1)
Here, A is a polynomial and Mn, Nn are 2 × 2 matrices whose entries are
polynomials [1]. In such a setting, in the present paper we study the Christof-fel, Geronimus, and Uvarov transformations for Laguerre-Hahn orthogonal polynomials within the modifications of systems of type (1). The main goal is to study the derivatives of the system (1) under rational modifications of the orthogonality weight and, as a consequence, to analyse the correspond-ing modifications on the three-term recurrence relation coefficients of the polynomials.
The reminder of the paper is organized as follows. In Section 2 we give the basic results on Laguerre-Hahn orthogonal polynomials and we introduce notation to be used in the sequel. In Section 3 we deduce formulae in the matrix form, involving the modified polynomials in terms of the original system. The modifications in the Laguerre-Hahn class are studied Section 4. The semi-classical class is analysed in Section 4.2.
2. Preliminary results and notations
Let P = span {xk : k ∈ N0} be the linear space of polynomials with complex
coefficients, and let P∗ be its algebraic dual space. We will denote by hu, f i the action of u ∈ P∗ on f ∈ P. Given the moments of u, un = hu, xni, n ≥ 0,
where we take u0 = 1, the principal minors of the corresponding Hankel
matrix are defined by Hn = det(ui+j)ni,j=0, where, by convention, H−1 = 1.
The functional u is said to be quasi-definite (respect., positive-definite) if Hn 6= 0 (respect., Hn > 0), for all n ≥ 0.
Let u ∈ P∗and let {Pn}n≥0be a sequence of polynomials such that deg(Pn) =
respect to u if
hu, PnPmi = hnδn,m, hn 6= 0 , n, m ≥ 0 . (2)
Throughout the paper we shall take each Pn monic, that is, Pn(x) = xn+
lower degree terms, and we will denote {Pn}n≥0 by SMOP.
The equivalence between the quasi-definiteness of u and the existence of a SMOP with respect to u is well-known in the literature of orthogonal polynomials [4, 18]. Furthermore, if u is positive-definite, then it has an integral representation in terms of a positive Borel measure, µ, supported on an infinite point set, I ⊆ R, such that
un = hu, xni =
Z
I
xndµ(x) , n ≥ 0 , (3)
and the orthogonality condition (2) becomes Z
I
Pn(x)Pm(x)dµ(x) = hnδn,m, hn > 0 , n, m ≥ 0 .
If µ is an absolutely continuous measure supported on I, and w denotes its Radon-Nikodym derivative with respect to the Lebesgue measure, then we will also say that {Pn}n≥0 is orthogonal with respect to w.
Associated with any SMOP there exist sequences {γn}n≥0 and {βn}n≥0 of
positive real numbers and real numbers, respectively, such that the three term recurrence relation holds [18]:
Pn+1(x) = (x − βn)Pn(x) − γnPn−1(x) , n ≥ 1 , (4)
with P0(x) = 1 and P1(x) = x − β0.
Given a SMOP with respect to u, the sequence of associated polynomials of the first kind is defined by
Pn(1)(x) = hut,
Pn+1(x) − Pn+1(t)
x − t i , n ≥ 0 ,
where ut denotes the action of u on the variable t.
The sequence {Pn(1)}n≥0 also satisfies a three-term recurrence relation,
Pn(1)(x) = (x − βn)P (1) n−1(x) − γnP (1) n−2(x) , n ≥ 1 , (5) with P−1(1)(x) = 0 and P0(1)(x) = 1.
The Stieltjes function of u is defined by S(x) = ∞ X n=0 un xn+1 . Note that if u is
positive-definite, defined by (3), then S is given by S(x) =
Z
I
dµ(t)
x − t , x ∈ C \ I .
The sequence of functions of the second kind corresponding to {Pn}n≥0 is
defined as follows:
qn+1 = Pn+1S − Pn(1), n ≥ 0 , q0 = S .
Whenever u is positive-definite, defined by (3), the qn’s are given in terms of
an integral formula, qn(x) = Z I Pn(t) x − tdµ(t) , n ≥ 0 .
The Stieltjes function S is said to be Laguerre-Hahn if there exist poly-nomials A, B, C, D, with A 6= 0, such that it satisfies a Riccati differential equation [15]
AS0 = BS2 + CS + D . (6)
The corresponding sequence of orthogonal polynomials is called Laguerre-Hahn. If B = 0, then S is said to be Laguerre-Hahn affine or semi-classical. Note that equation (6) is equivalent to the distributional equation [7, 15]
D(Au) = ψu + B(x−1u2) , ψ = A0 + C .
Furthermore, if u is positive-definite, defined in terms of a weight, w, then the semi-classical character of u, D(Au) = ψu, deg(ψ) ≥ 1, is equivalent to w0/w = C/A with w satisfying the boundary conditions [15]
xnA(x)w(x)|a,b = 0 , n ≥ 0 ,
where a, b (eventually a and/or b infinite) are linked with the roots of A. In such a case, w is the weight function on the support I = [a, b].
Throughout the paper we will use the following matrices associated with the SMOP {Pn}n≥0: Ψn = h Pn+1 Pn(1) iT , Yn = Pn+1 qn+1/w Pn qn/w , n ≥ 0 . (7)
In the account of (4) and (5), we have the recurrence relations Ψn(x) = (x − βn)Ψn−1(x) − γnΨn−2(x) , n ≥ 1 , (8) Yn = AnYn−1, An = x − βn −γn 1 0 , n ≥ 1 , (9)
with initial conditions Ψ−1 = h P0 P (1) −1 iT , Ψ0 = h P1 P (1) 0 iT , Y0 = x − β0 1 1 0 . As usual, An is called the transfer matrix.
The matrices (7) will play a relevant role in the sequel. Indeed, Laguerre-Hahn orthogonal polynomials are characterized in terms of differential sys-tems for {Ψn}n≥0: there holds the equivalence between (6) and
AΨ0n = MnΨn+ NnΨn−1, n ≥ 0 . (10)
Here, A is the same as in (6) and Mn, Nn are 2 × 2 matrices whose entries
are bounded degree polynomials depending on the coefficients of the Riccati equation [1, Theorem 1]. In the semi-classical case, that is, B ≡ 0 in (6), when dealing with weights, there holds the equivalence between the semi-classical character of w, w0/w = C/A, and the differential system
AYn0 = BnYn, n ≥ 1 . (11)
Here, Bn is a 2 × 2 matrix whose entries are bounded degree polynomials
depending on the polynomials A, C (see [2, Theorem 2] and [12]).
The three canonical transformations, the Christoffel, Geronimus and Uvarov transforms, are defined as follows [22].
(a) Christoffel: the weight is modified as ˜
w(x) = (x − α)w(x) , (12)
the polynomial modification is
(x − α) ˜Pn(x) = Pn+1(x) − anPn(x) , an = Pn+1(α)/Pn(α) . (13)
(b) Geronimus: the weight is modified as ˜
w(x) = w(x)
x − α, (14)
the polynomial modification is ˜
(c) Uvarov: the weight is modified as ˜
w(x) = w(x) + δα, (16)
the polynomial modification is
(x − α) ˜Pn(x) = Pn+1(x) + cnPn(x) + dnPn−1(x) , (17) with cn = α − βn + Pn(α)Pn−1(α) 1 + Kn−1(α, α) , dn = γn+ Pn2(α) 1 + Kn−1(α, α) . Here, Kn is defined by Kn(x, y) = n X k=0 Pk(x)Pk(y) hk , hk = k Y i=1 γi.
Throughout the text, (X)(i,j) will denote the (i, j) entry in the matrix X.
3. The Christoffel, Geronimus, and Uvarov
transforma-tions: matrix relations
In this section we will write the matrix modifications for {Ψn}n≥0 under
the Christoffel, Geronimus, and Uvarov transformations. 3.1. Christoffel transformation.
Theorem 1. Let {Pn}n≥0 be a SMOP related to a weight w and let { ˜Pn}n≥0
be the SMOP related to the modified weight (12). Denote by {Ψn}n≥0 and
{ ˜Ψn}n≥0 the corresponding sequences defined through (7). The following
re-lation holds: (x − α) eΨn(x) = Sn(x; α)Ψn+1(x) + Tn(x; α)Ψn(x) , n ≥ 0 , (18) where Sn(x; α) = 1 0 −1 x − α , Tn(x; α) = −an+1Sn(x; α) . (19)
Proof : Let us first prove the identity (x − α) gPn(1)(x) = an+1Pn+1(x) − Pn+2(x) + (x − α) Pn+1(1) (x) − an+1Pn(1)(x) . (20)
Using the definition of gPn(1), we get (x − α) gPn(1)(x) = (x − α) Z I ˜ Pn+1(x) − ˜Pn+1(t) x − t (t − α)w(t)dt . The use of (13) in the equation above gives us
(x − α) gPn(1)(x) =
Z
I
(t − α)An(x) − (x − α)An(t)
x − t w(t)dt ,
with An(x) = Pn+2(x) − an+1Pn+1(x). Thus, we get
(x − α) gPn(1)(x) = Z I tAn(x) − xAn(t) x − t w(t)dt − α Z I An(x) − An(t) x − t w(t)dt . Note that Z I tAn(x) − xAn(t) x − t w(t)dt = Z I (t − x)An(x) − (x − t)An(t) x − t w(t)dt + Z I xAn(x) − tAn(t) x − t w(t)dt , thus, Z I tAn(x) − xAn(t) x − t w(t)dt = −An(x) + xP (1) n+1(x) − an+1xPn(1)(x) , (21)
where we have used Z
I
(−An(x) − An(t))w(t)dt = −An(x) together with the
recurrence relation (4). Also, Z I An(x) − An(t) x − t w(t)dt = P (1) n+1(x) − an+1Pn(1)(x) . (22)
Therefore, (21) and (22) yield (20).
Equation (18) is the matrix form of (13) and (20).
In what follows we will see how the recurrence coefficients of the trans-formed orthogonal polynomials can be obtained through the matrix relations (18).
Corollary 1. Under the notation of (13), the recurrence coefficients of the SMOP { ˜Pn}n≥0 related to (12) are transformed according to the formulas
˜
βn = βn+1 + an+1− an, γ˜n = γn
an
an−1
Proof : Multiply the recurrence relation e
Ψn = (x − ˜βn) eΨn−1− ˜γnΨen−2, n ≥ 1 , by (x − α) and use (18), thus obtaining
SnΨn+1+ TnΨn = (x − ˜βn) (Sn−1Ψn+ Tn−1Ψn−1)
− ˜γn(Sn−2Ψn−1 + Tn−2Ψn−2) .
The use of the recurrence relation (8) in the above equality yields
Mn,1Ψn = Mn,2Ψn−1, n ≥ 0 , (24) where Mn,1 = (x − βn+1)Sn + Tn− (x − ˜βn)Sn−1− ˜ γn γn Tn−2, (25) Mn,2 = γn+1Sn+ (x − ˜βn)Tn−1 − ˜γnSn−2 − ˜ γn γn (x − βn)Tn−2. (26)
Taking into account (19), then (Mn,1)(1,2) = (Mn,2)(1,2) = 0. Hence, (24)
yields
(Mn,1)(1,1)Pn+1 = (Mn,2)(1,1)Pn.
As Pn and Pn+1do not share zeros (see, e.g., [4]) and the degrees of (Mn,j)(1,1),
j = 1, 2, are uniformly bounded, there follows
(Mn,1)(1,1) = (Mn,2)(1,1) = 0 .
Furthermore, in the account of (19), we have (Mn,1)(2,1) = − (Mn,1)(1,1),
(Mn,1)(2,2) = (x − α) (Mn,1)(1,1). Thus, Mn,1 is the null matrix. Therefore,
(24) reads (Mn,2)(2,1)Pn = − (Mn,2)(2,2)P (1) n−1. As Pn and P (1)
n−1do not share zeros (see, e.g., [4]) and the degrees of (Mn,2)(2,j),
j = 1, 2, are uniformly bounded, there follows
(Mn,2)(2,1) = (Mn,2)(2,2) = 0 ,
that is, Mn,2 is also a null matrix. Therefore, we get
(x − βn+1)Sn+ Tn − (x − ˜βn)Sn−1− ˜ γn γn Tn−2 = 0 , (27) γn+1Sn + (x − ˜βn)Tn−1 − ˜γnSn−2− ˜ γn γn (x − βn)Tn−2 = 0 . (28)
Equation (27) yields only one non-trivial equation, ˜ βn = βn+1 + an+1 − ˜ γn γn an−1. (29)
Equation (28) yields only one non-trivial equation, from which we get an =
˜ γn
γn
an−1. (30)
Equations (29) and (30) yield (23). 3.2. Geronimus transformation.
Theorem 2. Let {Pn}n≥0 be a SMOP related to a weight w and let { ˜Pn}n≥0
be the SMOP related to the modified weight (14). Denote by {Ψn}n≥0 and
{ eΨn}n≥0 the corresponding sequences defined through (7). The following
re-lation holds: (x − α) eΨn(x) = Sn(x; α)Ψn+1(x) + Tn(x; α)Ψn(x) , n ≥ 0 , (31) where Sn(x; α) = " 1 − (α− ˜βn+1)bn+1 γn+1 − ˜ γn+1 γn+1 + (x−βn)˜γn+1bn γnγn+1 0 bn+1 γn+1 bn+1 γn+1 # , (32) Tn(x; α) = " tn 0 1 − (x−βn+1)bn+1 γn+1 1 − (x−βn+1)bn+1 γn+1 # , (33) where tn = −bn+2 + ˜βn+1− α + ˜ γn+1bn γn − (x − βn+1)((Sn)(1,1) − 1) .
Proof : Let us first prove the identity
(x − α) gPn(1)(x) = Pn+1(x) − bn+1Pn(x) + Pn(1)(x) − bn+1P (1)
n−1(x) . (34)
Using the definition of gPn(1), we get
(x − α) gPn(1)(x) = (x − α) Z I ˜ Pn+1(x) − ˜Pn+1(t) x − t w(t) t − αdt = Z I ˜ Pn+1(x) − ˜Pn+1(t) t − α w(t)dt + Z I ˜ Pn+1(x) − ˜Pn+1(t) x − t w(t)dt . (35)
Note the first integral in (35), Z I ˜ Pn+1(x) − ˜Pn+1(t) t − α w(t)dt = ( ˜Pn+1(x) − ˜Pn+1(α)) Z I w(t) t − αdt − Z I ˜ Pn+1(t) − ˜Pn+1(α) t − α w(t)dt = − (Pn+1(x) − bn+1Pn(x) − Pn+1(α) + bn+1Pn(α)) S(α) − Pn(1)(α) + bn+1P (1) n−1(α) , (36)
where we have used (15).
The second integral in (35) gives us Z I ˜ Pn+1(x) − ˜Pn+1(t) x − t w(t)dt = P (1) n (x) − bn+1P (1) n−1(x) . (37)
Therefore, (36) and (37) yield
(x − α) gPn(1)(x) = − (Pn+1(x) − bn+1Pn(x) − Pn+1(α) + bn+1Pn(α)) S(α) −Pn(1)(α) + bn+1P (1) n−1(α) + P (1) n (x) − bn+1P (1) n−1(x) .
As gPn(1) is monic, there must hold −S(α) = 1, thus,
(x − α) gPn(1)(x) = Pn+1(x) − bn+1Pn(x) + Pn(1)(x) − bn+1P (1) n−1(x) + An(α) , with An(α) = −Pn+1(α) + bn+1Pn(α) − P (1) n (α) + bn+1P (1) n−1(α).
Note that An(α) = 0, hence, we obtain (34).
Now let us write, in the matrix notation, (15) and (34), that is, (x − α) eΨn = AΨn+1+ BnΨn+ CnΨn−1+ DnΨn−2, with A = 1 0 0 0 , Bn = ˜ βn+1 − bn+2 − α 0 1 1 , Cn = (α − ˜βn+1)bn+1+ ˜γn+1 0 −bn+1 − bn+1 , Dn = −˜γn+1bn 0 0 0 . The use of the recurrence relation (8) yields
with Sn = A − 1 γn+1 Cn − (x − βn) γnγn+1 Dn, Tn = Bn + (x − βn+1) γn+1 Cn + (x − βn)(x − βn+1) γnγn+1 − 1 γn Dn.
Hence, we obtain (31) with the matrices Sn, Tn given in (32) and (33).
Corollary 2. Under the notation of (15), the recurrence coefficients of the SMOP { ˜Pn}n≥0 related to (14) are transformed according to the formulas
˜
βn = βn + bn+1− bn, γ˜n = γn−1
bn
bn−1
. (38)
Proof : Proceeding as in the proof of Corollary 1 we obtain (24), Mn,1Ψn = Mn,2Ψn−1, n ≥ 0 ,
with Mn,1, Mn,2 given by (25) and (26), respectively.
Note that (Mn,1)(1,2) = (Mn,2)(1,2) = 0. Hence, following the same
argu-ments as in the proof of Corollary 1, there follows (Mn,1)(1,1) = (Mn,2)(1,1) = 0 .
Furthermore, as (Mn,1)(2,1) = (Mn,1)(2,2) and (Mn,2)(2,1) = (Mn,2)(2,2), after
some computations we obtain that Mn,1 is the null matrix and, consequently,
Mn,2 must also be null. Therefore, we have (27)–(28).
From position (2, 1) in (27) we get the equation for the ˜γn’s in (38),
˜ γn = γn−1 bn bn−1 , (39) as well as 1 + ˜ βn γn bn − ˜ γn γn 1 + bn−1 γn−1 bn−1 = 0 , (40)
thus, after some computations following from the use of (39) in (40), we get ˜ βn = βn−1 + γn−1 bn−1 − γn bn . (41)
From position (2, 1) in (28) we get βn−1 + γn−1 bn−1 − γn bn = βn+ bn+1− bn. (42)
3.3. Uvarov transformation.
Theorem 3. Let {Pn}n≥0 be a SMOP related to a weight w and let { ˜Pn}n≥0
be the SMOP related to the modified weight (16). Denote by {Ψn}n≥0 and
{ eΨn}n≥0 the corresponding sequences defined through (7). The following
re-lation holds: (x − α)2Ψen(x) = Sn(x; α)Ψn+1(x) + Tn(x; α)Ψn(x) , n ≥ 0 , (43) where Sn(x; α) = 1 − dn+1 γn+1 x − α 0 1 x − α , (44) Tn(x; α) = cn+1 + (x − βn+1) γn+1 dn+1 x − α 0 1 x − α . (45)
Proof : Let us first prove the identity
(x − α)2Pg(1)
n (x) = (x − α)Pn+1(1) (x) + (x − α)cn+1Pn(1)(x) + (x − α)dn+1P (1) n−1(x)
+ Pn+2(x) + cn+1Pn+1(x) + dn+1Pn(x) . (46)
Using the definition of gPn(1), we get
(x − α) gPn(1)(x) = Z I (x − α) ˜Pn+1(x) − (x − α) ˜Pn+1(t) x − t w(t)dt˜ = Z I (x − α) ˜Pn+1(x) − (t − α) ˜Pn+1(t) x − t w(t)dt˜ − Z I ˜ Pn+1(t) ˜w(t)dt . Note that Z I ˜ Pn+1(t) ˜w(t)dt = 0 . (47)
Also, Z I (x − α) ˜Pn+1(x) − (t − α) ˜Pn+1(t) x − t ˜ w(t)dt = Z I (x − α) ˜Pn+1(x) − (t − α) ˜Pn+1(t) x − t w(t)dt + Z I (x − α) ˜Pn+1(x) − (t − α) ˜Pn+1(t) x − t δα(t) . (48)
In account of (17), for the first integral in (48) we have Z I (x − α) ˜Pn+1(x) − (t − α) ˜Pn+1(t) x − t w(t)dt = Pn+1(1) (x) + cn+1Pn(1)(x) + dn+1P (1) n−1(x) .
For the second integral in (48) we have Z
I
(x − α) ˜Pn+1(x) − (t − α) ˜Pn+1(t)
x − t δα(t) = ˜Pn+1(x) . Hence, we obtain (46).
Now let us write, in the matrix form, (17) and (46). We get (x − α)2Ψen = AΨn+1 + BnΨn+ CnΨn−1, with A = x − α 0 1 x − α , Bn = cn+1A, Cn = dn+1A .
The use of the recurrence relation (8) yields
(x − α)2Ψen = SnΨn+1+ TnΨn, with Sn = A − 1 γn+1 Cn, Tn = Bn+ (x − βn+1) γn+1 Cn.
Hence, we obtain (43) with the matrices Sn, Tn given in (44) and (45).
Corollary 3. Under the notation of (17), the recurrence coefficients of the SMOP { ˜Pn}n≥0 related to (16) are transformed according to the formulas
˜
βn = βn+1 + cn − cn+1, γ˜n = γn−1
dn
dn−1
Proof : Proceeding as in the proof of Corollary 1 and Corollary 2 we obtain equations (27)–(28). From position (1, 1) in (27) we get the equation for the ˜
γn’s in (49), and after some computations we obtain, from (28), the equation
for the ˜βn’s.
3.4. Iterations - General formulas. We denote the iterated Christoffel transformation by w[K](x) = K Y j=1 (x − αj)w(x) , (50)
and the iterated Geronimus transformation by w[L](x) =
1 QL
j=1(x − νj)
w(x) . (51)
The monic orthogonal polynomials related to (50) and (51) will be denoted by Pn,K,· and Pn,·,L, respectively. The corresponding vectors defined in (7)
related to w[K] will be denoted by Ψn,K,·, and the ones related to w[L] will be
denoted by Ψn,·,L. Recall that the vectors related to w are denoted by Ψn.
Theorem 4. For the modification of the weight (50), the following relation holds, for all n ≥ 0:
K
Y
j=1
(x − αj)Ψn,K,·(x) = Sn(x; αk, . . . , α1)Ψn+1(x) + Tn(x; αk, . . . , α1)Ψn(x) ,
(52) where the matrices Sn(x; αk, . . . , α1), Tn(x; αk, . . . , α1) are defined recursively
through
Sn(x; αk, . . . , α1) = (x − βn+2)Sn(x; αk)Sn+1(x; αk−1, . . . α1)
+Sn(x; αk)Tn+1(x; αk−1, . . . , α1) + Tn(x; αk)Sn(x; αk−1, . . . , α1),
Tn(x; αk, . . . , α1) = −γn+2Sn(x; αk)Sn+1(x; αk−1, . . . , α1)
+Tn(x; αk)Tn(x; αk−1, . . . , α1) ,
with initial conditions Sn(x; α1) =
1 0
−1 x − α1
For the modification of the weight (51), the following relation holds: L Y j=1 (x − νj)Ψn,·,L(x) = Sn(x; νL, . . . , ν1)Ψn+1(x) + Tn(x; νL, . . . , ν1)Ψn(x) , (53) where the matrices Sn(x; νL, . . . , ν1), Tn(x; νL, . . . , ν1) are defined recursively
through
Sn(x; νL, . . . , ν1) = (x − βn+2)Sn(x; νL)Sn+1(x; νL−1, . . . , ν1)
+ Sn(x; νL)Tn+1(x; νL−1, . . . , ν1) + Tn(x; νL)Sn(x; νL−1, . . . , ν1),
Tn(x; νL, . . . , ν1) = −γn+2Sn(x; νL)Sn+1(x; νL−1, . . . , ν1)
+Tn(x; νL)Tn(x; νL−1, . . . , ν1) ,
with initial conditions Sn(x; ν1) = " 1 − (ν1− ˜βn+1)bn+1 γn+1 − ˜ γn+1 γn+1 + (x−βn)˜γn+1bn γnγn+1 0 bn+1 γn+1 bn+1 γn+1 # , Tn(x; ν1) = " tn 0 1 − (x−βn+1)bn+1 γn+1 1 − (x−βn+1)bn+1 γn+1 # , where tn = −bn+2 + ˜βn+1 − ν1 + ˜ γn+1bn γn − (x − βn+1)((Sn(x; ν1))(1,1)− 1) .
4. Modifications within the Laguerre-Hahn class
4.1. The Laguerre-Hahn class: the Christoffel, Geronimus, and Uvarov transformations. It is well known that the Laguerre-Hahn class is closed under the Christoffel, Geronimus, and Uvarov transformations [13,14]. Recall that Laguerre-Hahn orthogonal polynomials are characterized in terms of a Riccati equation for the Stieltjes function,
AS0 = BS2 + CS + D ,
or, equivalently , through a structure relation (10) for the corresponding Ψn,
AΨ0n = MnΨn+ NnΨn−1, n ≥ 0 ,
where Mn, Nn are matrices whose entries are bounded degree polynomials
Hence, there holds the equivalence between the Riccati equation for a mod-ification of the Stieltjes function under the christoffel, Geronimus, Uvarov transformations, say, ˜S, such that
˜
A ˜S0 = ˜B ˜S2 + ˜C ˜S + ˜D , and a structure relation for the corresponding eΨn,
˜
A eΨ0n = ˜MnΨ˜n+ ˜NnΨen−1, n ≥ 0 . (54) In order to construct the difference-differential equation (54) for the trans-formed eΨn, it is sufficient to know the difference-differential equation (10)
for Ψn and the matrices Sn, Tn which result from the
Christoffel-Geronimus-Uvarov transformations (cf. (18), (31), (43)), say, π eΨn = SnΨn+1+ TnΨn.
Indeed, we have the following result.
Theorem 5. Let {Pn}n≥0 be a SMOP related to a Laguerre-Hahn Stieltjes
function S satisfying AS0 = BS2 + CS + D, equivalently, the corresponding {Ψn}n≥0 satisfying the difference-differential equation (10),
AΨ0n = MnΨn+ NnΨn−1, n ≥ 0 .
Let { ˜Pn}n≥0 be the SMOP related to the modified Stieltjes function under the
Christoffel or Geronimus transformations (12) or (14), say ˜S satisfying ˜
A ˜S0 = ˜B ˜S2 + ˜C ˜S + ˜D . (55) Set the relations of the type given in (18), (31) written as
π eΨn = SnΨn+1+ TnΨn. (56)
There holds the equivalence between (55) and the difference-differential equa-tion
˜
A eΨ0n = ˜MnΨen + ˜NnΨen−1 (57) with ˜A = Aπ, and the matrices ˜Mn, ˜Nn given by
˜ Mn = b Sn+ 1 γn+1 ˜ NnTn−1 Sn−1, N˜n = b TnTn−1 − bSnSn−1 Vn−1, (58)
where b Sn = π ASn0 + SnMn+1 − 1 γn+1 TnNn − Aπ0Sn, (59) b Tn = π ATn0+ SnNn+1 + Tn Mn + (x − βn+1) γn+1 Nn − Aπ0Tn, (60) Vn = 1 γn+1 Tn−1Sn−1 + Sn−1+ (x − βn+1) γn+1 Tn−1 Tn−1. (61) Proof : Take derivatives in (56), multiply by the polynomial A, and use (10), thus obtaining Aπ0Ψen+ Aπ eΨ0n = Sn,1Ψn+1 + Tn,1Ψn, (62) with Sn,1 = ASn0 + SnMn+1− 1 γn+1 TnNn, Tn,1 = ATn0 + SnNn+1 + Tn Mn+ (x − βn+1) γn+1 Nn . Multiply (62) by π and use again (56), thus obtaining
ˆ
A eΨ0n = bSnΨn+1 + bTnΨn, (63)
with
ˆ
A = Aπ2, Sbn = πSn,1− Aπ0Sn, bTn = πTn,1− Aπ0Tn.
Let us now deduce (58). We consider the relation (57) with ˜A = Aπ, that is,
Aπ eΨ0n = ˜MnΨen + ˜NnΨen−1.
Multiplying the above equation by π and using (56) and (63), as well as the recurrence relation (8), we get
Mn,1Ψn+1 = Mn,2Ψn, with Mn,1 = bSn − ˜MnSn+ 1 γn+1 ˜ NnTn−1, (64) Mn,2 = − bTn + ˜MnTn + ˜NnSn−1 + (x − βn+1) γn+1 ˜ NnTn−1. (65)
Taking into account the structure of the matrices defining Mn,1 and Mn,2,
we conclude that Mn,1 and Mn,2 are null matrices. Hence, we obtain that
˜
Mn, ˜Nn are given by (58).
4.2. The semi-classical class: the Christoffel and Geronimus trans-formations. The semi-classical class is closed under the Christoffel and Geronimus transformations [15].
Recall that, whenever dealing with weights, there holds the equivalence be-tween the semi-classical character of w, say w0/w = C/A, and the differential system (11) for the corresponding Yn,
AYn0 = BnYn, n ≥ 1 ,
where Bn is a matrix of polynomial entries having uniformly bounded degrees
[2, Theorem 2].
Let us now consider ˜w, a modification of w under the Christoffel or Geron-imus transformation. As ˜w is semi-classical, there holds a differential system for the corresponding ˜Yn =
˜ Pn+1 q˜n+1/ ˜w ˜ Pn q˜n/ ˜w , ˜ A ˜Yn0 = ˜BnY˜n, n ≥ 1 . (66)
In what follows we see how ˜Bn in (66) and the transfer matrices of { ˜Yn}n≥0,
henceforth denoted by ˜An, relate to Bn in (11) and to the transfer matrices
An of {Yn}n≥0.
Theorem 6. Let {Pn}n≥0 be a SMOP related to a semi-classical weight w
with the corresponding {Yn}n≥0 satisfying (11),
AYn0 = BnYn, n ≥ 1 .
Let An be the transfer matrix of Yn.
a) Christoffel case.
a.1) The sequence { ˜Yn}n≥0 related to the modified weight (12), ˜w(x) = (x −
α)w(x), satisfies (x − α) ˜Yn = CnYn, Cn = An+1 − An, n ≥ 1 , (67) where An = an+1 0 0 an , an = Pn+1(α)/Pn(α);
a.2) { ˜Yn}n≥0 satisfies a linear system (66), ˜A ˜Yn0 = ˜BnY˜n, n ≥ 1 , where
˜
a.3) The transfer matrices of ˜Yn and Yn are related through
˜
An = CnAnCn−1−1 , n ≥ 1 . (69)
b) Geronimus case.
b.1) The sequence { ˜Yn}n≥0 related to the modified weight (14), ˜w(x) =
w(x) (x − α), satisfies ˜ Yn = GnYn, Gn = An+1 − Bn, (70) where Bn = bn+1 0 0 bn , bn = qn(α)/qn−1(α);
b.2) { ˜Yn}n≥0 satisfies a linear system (66), ˜A ˜Yn0 = ˜BnY˜n, n ≥ 1 , where
˜
A = (x − α)A , B˜n = (x − α) (AGn0 + GnBn) Gn−1; (71)
b.3) The transfer matrices of ˜Yn and Yn are related through
˜
An = GnAnGn−1−1 . (72)
Proof : Christoffel case.
a.1) The sequence of functions of the second kind related to ˜w, {˜qn}, satisfies
˜
qn(x) = qn+1(x) − anqn(x) , n ≥ 1 . (73)
Thus,
(x − α) ˜Yn = Yn+1− AnYn.
The use of (9), Yn+1 = An+1Yn, in the equation above yields (67).
a.2) Take derivatives in (67), then multiply the resulting equation by (x − α)A and use (67) again, thus obtaining
(ACn + ˜BnCn)Yn = (x − α) (ACn0 + CnBn) Yn.
Hence, we have the linear system (66) with the data (68).
a.3) Relation (69) follows from the use of the recurrence relation for Yn as
well as for ˜Yn, into (67).
Geronimus case.
b.1) The sequence of functions of the second kind related to ˜w, {˜qn}, satisfies
(x − α)˜qn(x) = qn(x) − bnqn−1(x) , n ≥ 1 .
Thus,
˜
Yn = Yn+1 − BnYn.
b.2) Take derivatives in (70), then multiply the resulting equation by (x − α)A and use (70) again, thus obtaining
˜
BnGnYn = (x − α) (AGn0 + GnBn) Yn.
Hence, we have the linear system (66) with the data (71).
b.3) Relation (72) follows from the use of the recurrence relation for Yn as
well as for ˜Yn into (70).
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K. Castillo
CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal.
E-mail address: kenier@mat.uc.pt M.N. Rebocho
Department of Mathematics, University of Beira Interior, 6201-001 Covilh˜a, Portugal; CMUC, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal.
